Characterization of Upper Comonotonicity via Tail Convex Order

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1 Characterization of Upper Comonotonicity via Tail Convex Order Hee Seok Nam a,, Qihe Tang a, Fan Yang b a Department of Statistics and Actuarial Science, University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, USA b Applied Mathematical and Computational Sciences, University of Iowa, Iowa City, IA 52242, USA Abstract In this paper we show a characterization of upper comonotonicity via tail convex order. For any given marginal distributions, a maximal random vector with respect to tail convex order is proved to be upper comonotonic under suitable conditions. As an application, we consider the computation of the Haezendonck risk measure of the sum of upper comonotonic random variables with exponential marginal distributions. Keywords: comonotonicity, upper comonotonicity, tail convex order, Haezendonck risk measures JEL: D81; G32 Subject Category and Insurance Branch Category: IM30; IE43 1. Introduction In the study of the riskiness of an insurance portfolio, we are often interested in the total loss as a sum of risk variables. While traditional approaches have been based on the independence assumption between risks, recent trends allow dependent structures so that we can model the real situation more plausibly. Complications due to dependence between risks can be avoided by, or appropriately dealt with through, replacing the original sum by a less attractive one with a simpler dependence structure. Several Corresponding author addresses: kinsuever@gmail.com (Hee Seok Nam), qihe-tang@uiowa.edu (Qihe Tang), fan-yang-2@uiowa.edu (Fan Yang) Preprint submitted to Insurance: Mathematics and Economics January 5, 2011

2 concepts have been introduced to compare risks in the actuarial literature (see e.g. [8], [11]). One common way is to consider convex order which uses stop-loss premiums with the same mean. It is well known that if a random vector with given marginal distributions is comonotonic, then it has the largest sum with respect to convex order (see e.g. [9]). A proof based on geometric interpretation of the comonotonic support can be found in [15]. Conversely, it is also true that if a sum is maximal in the set of all random vectors with predetermined marginal distributions in convex order, then the underlying random variables must be comonotonic. Cheung proved the claim for the case of continuous marginal distributions in [3]. Later it was generalized to the case of integrable distributions in [5]. In [16] the authors showed that the problem boils down to a bivariate case and the joint distribution is indeed the minimum of the two marginal distributions. The well-known fact that a random vector is comonotonic if and only if it is pairwise comonotonic is used. It is, however, no longer valid for an upper comonotonic random vector and a counter-example is given in the appendix of this paper. The concept of upper comonotonicity was introduced and investigated in [4]. A characterization using joint distributions and the additivity of Valueat-Risk, Tail Value-at-Risk, and Expected Shortfall were given. In [12] the authors proved the additivity of α-mixed inverse distribution functions and stop-loss premiums for the sum of upper comonotonic random variables. In [5] it was shown that the additivity of Value-at-Risk for the confidence level being sufficiently large implies the upper comonotonicity. On the other hand, as in the comonotonic case, we can consider an ordering between the sum S of the components of a random vector X and the corresponding sum S uc of an upper comonotonic random vector X uc. Indeed, if X uc coincides with X in the lower tail in distribution and they have the same marginal distributions, then the sum S precedes S uc in convex order (for details, see [12]). If there is no coincidence in the lower tail, then the convex order is no longer valid in general and it should be relaxed to describe the upper tail only. The concept of tail convex order was introduced by [6], who proved that, upon suitable conditions, the sum of the components of an upper comonotonic random vector is the largest in tail convex order. This suggests that upper comonotonicity be related to tail convex order rather than convex order. Therefore, the following characterization question arises: if the sum of the components of a random vector is maximal in tail convex order, then is 2

3 this random vector upper comonotonic? In the present paper, we shall focus on this question and pursue an answer. The rest of this paper is organized as follows: Section 2 prepares some necessary definitions and states our main result, Section 3 formulates the proof of the result, Section 4 shows an application to the computation of Haezendonck risk measures, and, finally, Appendix contains some examples to further clarify some fallacies. 2. Main Result The underlying probability space is denoted by (Ω, F, P ), where real valued random variables or risks X i and Y i are defined for i = 1,..., n. A subset C R n is said to be comonotonic if for any x and y in C, either x y or y x holds. We call a random vector X = (X 1,..., X n ) to be comonotonic if it has a comonotonic support. As a generalization of comonotonicity, we consider upper comonotonicity. For any given d = (d 1,..., d n ) (R { }) n, the upper quadrant (d 1, ) (d n, ) is denoted by U(d), the lower quadrant (, d 1 ] (, d n ] by L(d), and the remaining complement R n \ (U(d) L(d)) by R(d). Definition 2.1. ([4]) A random vector X = (X 1,..., X n ) is said to be upper comonotonic if there exist some d (R { }) n and some null set N such that (i) {X(ω) : ω Ω \ N} U(d) is comonotonic, (ii) P (X U(d)) > 0, (iii) {X(ω) : ω Ω \ N} R(d) is empty. The usual inverse distribution functions F 1 1+ X (p), FX (p) of a random variable X : Ω R are defined by F 1 X (p) = inf{x R : F X(x) p}, F 1+ X (p) = sup{x R : F X(x) p}, respectively, with the convention inf = and sup =. To pick up any point in the closed interval [F 1 1+ X (p), FX (p)], we introduce the so-called α-mixed inverse distribution functions of F X as F 1(α) X (p) = αf 1 1+ (p) + (1 α)f (p), p (0, 1). X X 3

4 See e.g. [9] for related discussions. From now on, the distribution function F Xi or F Yi will be denoted by F i as long as no confusion arises. A random variable X is said to precede another random variable Y in stop-loss order (written as X sl Y ) if X has less stop-loss premiums than Y, i.e. E[(X d) + ] E[(Y d) + ] for all d R. It is well known that stop-loss order preserves the ordering of TVaR p and vice versa, i.e. X sl Y if and only if TVaR p [X] TVaR p [Y ] for all p (0, 1) (see [11]). A random variable X is said to precede Y in convex order (written as X cx Y ) if E[X] = E[Y ] and X sl Y. For an overview of recent progresses on comonotonicity-based convex bounds, see [7], [10] and references therein. In particular, by limiting the range of d in stop-loss order, we are led to the following definition. Definition 2.2. ([6]) A random variable X is said to precede another random variable Y in tail convex order (written as X tcx Y ) if there exists some d such that (i) P (X d ) > 0, (ii) E[X] = E[Y ], (iii) E[(X d) + ] E[(Y d) + ] for all d d. For any given Y, the existence of an upper comonotonic random vector Y uc satisfying S tcx S uc is shown in Section 4 of [12], where (here and throughout) S and S uc are the sums of the components of Y and Y uc, respectively. For any given distribution functions F 1,..., F n, the Fréchet space R = R(F 1,..., F n ) is defined to be the set of all random vectors with marginal distributions F 1,..., F n. Note that if we restrict ourselves on R, then stoploss order is exactly the same as convex order. Now we are ready to state the main result. In what follows, for any given A R n, its closure, interior, and boundary are denoted by A, int(a), and bd(a), respectively. Theorem 2.3. For any given marginal distributions F 1,..., F n, if there exists Y = (Y 1,..., Y n ) R = R(F 1,..., F n ) such that X X n tcx Y Y n, (X 1,..., X n ) R, (1) then there exist some d R n and a null set N Ω such that 4

5 (i) P (Y U(d )) > 0, (ii) {Y : ω Ω \ N} U(d ) is comonotonic, (iii) {Y : ω Ω \ N} int(r(d )) =. Furthermore, if P (Y bd(u(d ))) = 0 then (Y 1,..., Y n ) is upper comonotonic with a threshold in L(d ). Note that (i) (iii) of the theorem are similar to, but not exactly same as, the three conditions in Definition 2.1. Example A.2 below shows the necessity of the condition P (Y bd(u(d ))) = 0 for Y to be upper comonotonic. 3. Proof of the Theorem The proof of Theorem 2.3 will be given by observing relations between Y and its comonotonic counterpart Y c in terms of stop-loss premiums Lemmas To begin with, we recall the following lemma: Lemma 3.1. ([9]) The stop-loss premiums of the sum S c of the components of the comonotonic random vector Y c = (Y1 c,..., Yn c ) are given by E[(S c d) + ] = n E[(Yi c d i ) + ] for F 1+ 1 S (0) < d < F c S (1), c i=1 where d i = F 1(α) i (F S c(d)) and α [0, 1] solves the equation F 1(α) S (F c S c(d)) = d. Note that a similar result also holds true for an upper comonotonic random vector (see e.g. [12]). In addition, we can see that d i = F 1(α) i (q) is defined only for values of the form q = F S c(d). Nevertheless, the applicability of Lemma 3.1 during the proof of Theorem 2.3 is guaranteed by the following lemma, which shows that the image of each F i is a subset of the image of F S c, namely, Im(F i ) Im(F S c). Lemma 3.2. For any random vector Y = (Y 1,..., Y n ) and its comonotonic counterpart Y c, we have Im(F i ) Im(F S c), i = 1,..., n. 5

6 Proof. We prove the contraposition. It is well known that the additivity of the α-mixed inverse function holds for a comonotonic random vector [9]: F 1(α) S c (q) = n i=1 F 1(α) i (q), 0 < q < 1, 0 α 1. (2) If q / Im(F S c), then there exists a sufficiently small ɛ > 0 such that F 1(α) S c is constant on the interval [q, q + ɛ). By (2) each F 1(α) i is also constant on the interval [q, q + ɛ) since it is non-decreasing,. Then q / Im(F i ) for each i {1,..., n} and this completes the proof Proof of Theorem 2.3 The argument of [16] motivates the following proof. Since the comonotonic counterpart Y c is also in the Fréchet space R, the condition (1) implies Y c Y c n tcx Y Y n. By definition there exists d such that P (S d ) > 0 and E[(Y Y n d) + ] = E[(Y c Y c n d) + ], d d. (3) For d d and α [0, 1] which solves the equation F 1(α) S (F c Sc(d)) = d, we define d = (d 1,..., d n ) with d i = F 1(α) i (F S c(d)). In the particular case of d = d, the notations d and d i are used. Then, by (3) and Lemma 3.1, [ n n ] E[(S d) + ] = E[(S c d) + ] = E[(Yi c d i ) + ] = E (Y i d i ) +. i=1 Since (S d) + n i=1 (Y i d i ) +, we have (S d) + = n i=1 (Y i d i ) + almost surely, or, equivalently, there exists a null set N Ω such that (Y i (ω) d i )(Y j (ω) d j ) 0, ω Ω \ N, i, j = 1,..., n. (4) Since P (S d ) > 0, (4) implies i=1 P (Y U(d )) > 0. (5) We formulate the remaining proof of Theorem 2.3 into the following three steps. 6

7 Step 1. For any (y i, y j ) U(d i, d j ), the closure of the upper quadrant U(d i, d j), we have P (Y i y i, Y j y j ) = min k=i,j P (Y k y k ). Proof. Note that a distribution function has no more than countably many discontinuities. If we can show that, for any (y i, y j ) U(d i, d j) with y i and y j being continuities of F i and F j, there exists a null set Ñ Ω such that {ω Ω \ N : Y i y i, Y j y j } = {ω Ω \ Ñ : Y k y k }, k = i or j, then the right continuity of distribution functions gives the desired result of Step 1. Without loss of generality, we may assume that i = 1, j = 2, that F 1, F 2 are continuous at y 1, y 2, and that q 1 = F 1 (y 1 ) q 2 = F 2 (y 2 ). It suffices to prove that P (Y 1 y 1, Y 2 y 2 ) = P (Y 1 y 1 ). (6) Lemma 3.2 guarantees that q 1, q 2 Im(F S c) and we can use (4) to obtain ( )( ) Y 1 (ω) F 1(α 1) 1 (q 1 ) Y 2 (ω) F 1(α 1) 2 (q 1 ) ( )( ) 0, ω Ω \ N, (7) Y 1 (ω) F 1(α 2) 1 (q 2 ) Y 2 (ω) F 1(α 2) 2 (q 2 ) 0, ω Ω \ N, (8) where α 1 and α 2 [0, 1] solve the equations y 1 = F 1(α 1) 1 (q 1 ) and y 2 = F 1(α 2) 2 (q 2 ), respectively. If q 1 = F 1 (y 1 ) < q 2 = F 2 (y 2 ), then, by the continuity of F 1 at y 1 and (7), {ω Ω \ N : Y 1 y 1 } {ω Ω \ N : Y 1 y 1, Y 2 y 2 } = {ω Ω \ N : Y 1 y 1, Y 2 F 1(α 2) 2 (q 2 )} = {ω Ω \ N : Y 1 < y 1, Y 2 F 1(α 1) 2 (q 1 )} {ω Ω \ N : Y 1 = y 1, Y 2 F 1(α 1) 2 (q 1 )} {ω Ω \ N : Y 1 y 1, F 1(α 1) 2 (q 1 ) < Y 2 F 1(α 2) 2 (q 2 )} = {ω Ω \ N : Y 1 < y 1, Y 2 F 1(α 1) 2 (q 1 )} N 1 = {ω Ω \ N : Y 1 < y 1 } N 1 = {ω Ω \ N : Y 1 y 1 } N 1 \ N 2, (9) 7

8 where P (N 1 ) = P (N 2 ) = 0. If q 1 = q 2 and α 1 < α 2, then the same argument also applies to get (9). Finally, if q 1 = q 2 and α 1 α 2, then, by the continuity of F 2 at y 2 and (8), {ω Ω \ N : Y 2 y 2 } {ω Ω \ N : Y 1 y 1, Y 2 y 2 } = {ω Ω \ N : Y 1 F 1(α 1) 1 (q 1 ), Y 2 y 2 } = {ω Ω \ N : Y 1 F 1(α 2) 1 (q 2 ), Y 2 < y 2 } {ω Ω \ N : Y 1 F 1(α 2) 1 (q 2 ), Y 2 = y 2 } {ω Ω \ N : F 1(α 2) 1 (q 2 ) < Y 1 F 1(α 1) 1 (q 2 ), Y 2 y 2 } = {ω Ω \ N : Y 1 F 1(α 2) 1 (q 2 ), Y 2 < y 2 } N 3 = {ω Ω \ N : Y 2 < y 2 } N 3 = {ω Ω \ N : Y 2 y 2 } N 3 \ N 4, (10) where P (N 3 ) = P (N 4 ) = 0. By (9) and (10) we conclude that the relation (6) holds, and this completes the proof of Step 1. Step 2. For any y = (y 1,..., y n ) U(d ), and Y is comonotonic in U(d ). P (Y 1 y 1,..., Y n y n ) = min i=1,...,n P (Y i y i ), Proof. By induction, we will prove the claim that {ω Ω \ N : Y 1 y 1,..., Y n y n } = {ω Ω \ Ñ : Y i y i } (11) for some i {1,..., n} with null sets N and Ñ. For n = 2, the claim is true by Step 1. Now, suppose that the claim is true for n = k 2 and consider it for n = k + 1. Again by Step 1, we have {ω Ω \ N : Y 1 y 1,..., Y k+1 y k+1 } = {ω Ω \ N : Y 1 y 1,..., Y k y k } {ω Ω \ N : Y k+1 y k+1 } = {ω Ω \ N i : Y i y i } {ω Ω \ N : Y k+1 y k+1 } for some i {1,..., k} = {ω Ω \ (N N i ) : Y i y i, Y k+1 y k+1 } = {ω Ω \ Ñ : Y j y j } for j = i or k + 1, where N, N i, and Ñ are null sets. This shows that (11) holds true. 8

9 To show the comonotonicity of Y on U(d ), we construct a new random vector Ỹ by restricting Y to the set Ω = {ω Ω \ N : Y(ω) U(d )}. Then by (4), P ( Ω) = 1 F Y (d ) + P (Y = d ). Note that P ( Ω) > 0 by (5). Now we consider the new probability space ( Ω, F, P ) where F = F Ω and P (A) = P (A)/P ( Ω) for A F. To obtain the comonotonicity of Ỹ = Y Ω, it suffices to check FỸ(y 1,..., y n ) = min 1 i n F Ỹ i (y i ), (y 1,..., y n ) R n. (12) Since the interior of R(d ) has no intersection with {Y(ω) : ω Ω \ N} by (4), if y i d i then FỸi (y i ) = P ({Y i y i } Ω)/P ( Ω) = ( F i (y i ) (1 P ( Ω)) ) /P ( Ω). Therefore the marginal distribution function of Ỹi is given by { 0, y i < d i, FỸi (y i ) = ( Fi (y i ) (1 P ( Ω)) ) /P ( Ω), y i d i. Similarly, for any y R n we have { 0, y / U(d FỸ(y) = ), ( FY (y) (1 P ( Ω)) ) /P ( Ω), y U(d ). Finally, by (11) we obtain that (12) holds true and, hence, Y is comonotonic on U(d ). Step 3. If P (Y bd(u(d ))) = 0 then (Y 1,..., Y n ) is upper comonotonic with a threshold in L(d ). Proof. The condition of this step is equivalent to P (Y 1 d 1,..., Y i 1 d i 1, Y i = d i, Y i+1 d i+1,..., Y n d n) = 0 for each i {1,..., n}. In other words, (4) with a strict inequality holds, i.e., ( )( ) Y i (ω) d i Y j (ω) d j > 0, ω Ω \ N, i, j = 1,..., n. This implies that R(d ) {Y(ω) : ω Ω \ N} =. Then it is straightforward to check the conditions (i)-(iii) in Definition 2.1 to get the upper comonotonicity of Y. 9

10 4. Application In this section, we consider an application to the computation of Haezendonck risk measures. The concept of Haezendonck risk measures was first raised by Haezendonck and Goovaerts [14]. As a premium calculation principle induced by an Orlicz norm, it is multiplicatively analogous to the zero utility principle. In [13], Haezendonck risk measures were derived as translationinvariant minimal Orlicz risk measures which preserve convex order. In [1], an alternative formulation of Haezendonck risk measures was introduced to guarantee coherence. For application to portfolio optimization, several estimators of Haezendonck risk measure were provided by numerical simulations in [2]. Definition 4.1. ([13]) For a risk variable X, 0 < q < 1 and a nonnegative, strictly increasing continuous function φ( ) satisfying φ(0) = 0, φ(1) = 1, φ(+ ) = +, the Haezendonck risk measure is defined as π q [X] = inf π q[x, x], (13) <x<max[x] where π q [X, x] is the unique solution of the equation [ ( )] (X x)+ E φ = 1 q. (14) π x As its definition indicates, the computation of the Haezendonck risk measure π q [X] is complicated in general except for some ideal choices of φ( ) and the distribution of X. Example 4.2. Suppose that φ(t) = t. For any given marginal distributions F 1,..., F n, if Y = (Y 1,..., Y n ) R(F 1,..., F n ) is maximal in tail convex order, then there exists some q (0, 1) such that π q [Y Y n ] = n TVaR q [Y i ], q (q, 1). (15) i=1 Proof. By definition there exists some d such that E[(S c x) + ] = E[(S x) + ] for all x d. Thus π q [S, x] = π q [S c, x] for all x d since π q [S, x] = E[(S x) +] 1 q 10 + x.

11 Moreover, if x d then F S (x) = F S c(x) by (ii) and (iii) of Theorem 2.3. Indeed, the support of Y coincides with that of Y c on U(d ) \ d and both of Y and Y c have no support on int(r(d )). The value of the Haezendonck risk measure π q [S] in (13) is attained at x at which the derivative d dx π q[s, x] = F S(x) q 1 q changes sign. A subtle analysis shows that x = F 1 S (q). Therefore, if q > q := F S (d ) then and x = F 1 S 1 (q) = F c (q) d π q [S] = π q [S, x ] = π q [S c, x ] = E[(Sc F 1 S (q)) +] c + F 1 S 1 q (q). c Hence, the desired relation (15) holds. In fact, the proof of Example 4.2 shows that π q [Y Y n ] = TVaR q [Y Y n ], q (0, 1) with φ(t) = t (see e.g. [13]). For a more general φ of the form φ(t) = t r with r > 1, it is unlikely that we can establish a transparent expression for the Haezendonck risk measure π q [X]. Nevertheless, for some special cases we can do so under the help of the following lemma. Lemma 4.3. If φ(t) = t r with r 1 and 0 < q < 1 then d dx π q[x, x] is non-decreasing in x and lim x Proof. For φ(t) = t, as shown in Example 4.2, S d dx π q[x, x] = 1. (16) d dx π q[x, x] = F X(x) q, 1 q 11

12 which is non-decreasing and (16) holds true. For φ(t) = t r with r > 1, we have ( ) 1 d 1 dx π r ( q[x, x] = 1 E[(X x) r 1 q + ] ) 1 r 1 E[(X x) r 1 + ]. (17) Since r > 1, an application of Hölder s inequality shows that ( d 2 1 dx π q[x, x] = 2 Indeed, E[(X x) r 1 + ] = E 0. [ ) 1 r (r 1) ( E[(X x) r + ] ) 1 r 2 1 q ( (E[(X x) r 1 + ] ) 2 E[(X x) r + ]E[(X x) r 2 + ] (X x) r 2 + (X x) r Thus d dx π q[x, x] is continuous and non-decreasing. Finally, by Hölder s inequality again we have ] ( E[(X x) r +] ) 1 2 ( E[(X x) r 2 + ] ) 1 2. E[(X x) r 1 + ] = E [ (X x) r (x<x< ) ] ( E[(X x) r + ] ) 1 1 r (1 F X (x)) 1 r, which, together with (17), implies (16) and the proof ends. An upper comonotonic random vector Y = (Y 1,..., Y n ) has a stochastic representation (Y 1,..., Y n ) = d (F1 1 (U 1 ),..., Fn 1 (U n )), where U i = U1 (U>β) + βv i 1 (U β) for some β [0, 1), U and V i s are uniform on (0, 1), and U is independent of (V 1,..., V n ) (see e.g. [4]). Example 4.4. Suppose that φ(t) = t r with r 1. If the marginal distributions are exponential with rates λ 1,..., λ n, respectively, then ) π q [S] = 1 λ ln rr (1 q) Γ(r + 1) + r λ (18) for q > 1 Γ(r+1) (1 β) and λ = ( n 1 r r i=1 λ i ) 1. 12

13 Proof. We have E[(S x) r +] = E [ (S x) r +(1 (U>β) + 1 (U β) ) ] [( ) r ] [( n ln(1 U) = E x 1 (U>β) + βe λ + If λx ln(1 β) then it reduces to 1 ( ln(1 u) E[(S x) r +] = λ β ) r x + i=1 ln(1 βv i ) λ i x du = e λx Γ(r + 1), λr so that d dx π q[s, x] = 1 1 ( ) 1 Γ(r + 1) r e λ x r. r 1 q d By Lemma 4.3, if π dx q[s, x] < 0, or, equivalently, ln(1 β) x= q > 1 λ Γ(r + 1) (1 β), r r then the equation d dx π q[s, x] = 0 has a solution x = 1 λ ln rr (1 q) Γ(r + 1) ln(1 β) >. λ Thus we obtain the desired Haezendonck risk measure (18) as π q [S] = π q [S, x ]. Note that if β = 0, then this example reduces to the case of completely comonotonic random vector. 5. Conclusion The contribution of this paper is twofold. First, we showed a characterization of upper comonotonicity via tail convex order. Second, as an application, we considered the computation of the Haezendonck risk measure of the sum of upper comonotonic random variables and we established a transparent expression for the case with exponential marginal distributions and a large confidence level. Note that upper comonotonicity corresponds to a special case of asymptotic dependence in the upper tail with coefficient 1. It is desirable to extend some applications of upper comonotonicity to the case of asymptotic dependence. We shall pursue such extensions in our future research. 13 ) r + ].

14 Acknowledgments The authors are thankful to Ka Chun Cheung and an anonymous referee for their helpful comments on an earlier version of this paper. Appendix It is well known that a random vector Y = (Y 1,..., Y n ) is comonotonic if and only if it is pairwise comonotonic (see e.g. [9]). This is, however, no longer true for an upper comonotonic random vector. Example A.1. For n = 3 and Ω = {ω 1,..., ω 8 }, we define a random vector Y = (Y 1, Y 2, Y 3 ) as follows: ω Y 1 Y 2 Y 3 P ω /8 ω /8 ω /8 ω /8 ω /8 ω /8 ω /8 ω /8 It is easy to check that each pair (Y i, Y j ) is upper comonotonic. Indeed, the thresholds of (Y 1, Y 2 ), (Y 2, Y 3 ), (Y 3, Y 1 ) are (0, 0), ( 1, 1), and (0, 1), respectively. Note, however, that the comonotonicity is broken down due to ω 3 and ω 4. Since Y(ω i ) and Y(ω i+1 ) always have at least one component with the same values for i = 1,..., 7 like a chain, there is no point d R 3 satisfying the three conditions in Definition 2.1. Therefore Y is not upper comonotonic. The following example illustrates the necessity of the condition P (Y bd(u(d ))) = 0 in Theorem 2.3 for Y to be upper comonotonic. Example A.2. For n = 2 and Ω = {ω 1,..., ω 10 }, we define a random vector Y = (Y 1, Y 2 ) with the corresponding comonotonic counterpart (Y1 c, Y2 c ) as follows: 14

15 ω Y 1 Y 2 Y 1 + Y 2 P ω /10 ω /10 ω /10 ω /10 ω /10 ω /10 ω /10 ω /10 ω /10 ω /10 ω Y1 c Y2 c Y1 c + Y2 c P ω /10 ω /10 ω /10 ω /10 ω /10 ω /10 ω /10 ω /10 ω /10 ω /10 It is easy to check that the stop-loss premiums of Y 1 + Y 2 and Y c 1 + Y c 2 are equal for all d 0, i.e., E[(Y c 1 + Y c 2 d) + ] = E[(Y 1 + Y 2 d) + ], d 0. Although the support of (Y 1, Y 2 ) is comonotonic on U(0, 0), (Y 1, Y 2 ) is not upper comonotonic. The reason exists in that P ((Y 1, Y 2 ) bd(u(0, 0))) = 1 10 > 0. References [1] F. Bellini, E. Rosazza Gianin, On Haezendonck risk measures, Journal of Banking & Finance 32 (2008) [2] F. Bellini, E. Rosazza Gianin, Optimal portfolios with Haezendonck risk measures, Statist. Decisions 26 (2008) [3] K.C. Cheung, Characterization of comonotonicity using convex order, Insurance Math. Econom. 43 (2008) [4] K.C. Cheung, Upper comonotonicity, Insurance Math. Econom. 45 (2009) [5] K.C. Cheung, Characterizing a comonotonic random vector by the distribution of the sum of its components, Insurance Math. Econom. 47 (2010)

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